Your list of requirements will most certainly drive the price up and limit options. Here are some things to consider.OS
Depending on which type of live TV you wish to consume will determine with version of windows you can use. Windows 7 is the cheapest option for digital cable (when dealing with a provider that flag the majority of channels as copy once) Windows 8x also works but getting a windows 8 pro + media center license is expensive and microsoft no longer sells them (only 3rd party stock left at this point) Windows 10 I wouldn't recommend at all currently as the only copy once solution is with silicondust hdhomerun equipement and their feature lacking software.If you have a provider that don't flag everything as copy once or you only wish to watch over the air TV then os choice isn't as big of deal since other choices exist.
Hardware
By throwing light gaming into the mix I would suggest having no less than an Intel hd530 which is built into the following cpu:
G4500G4500TG4520i3-6300i3-6300Ti3-6320i3-6100i3-6100Ti5-6600i5-6600Ki5-6600Ti5-6500i5-6500Ti5-6400Ti5-6400i7-6700Ki7-6700Ti7-6700.
The G4500 should be the cheapest but it may be worth looking at one of the I3 models if you wish to view 3d framepacked blu-rays or play any of the games that refuse to start without a 4 threaded cpu. AMD apu's are an option but if cpu bound games such as mmorpgs are part of the plan then I would avoid them. I would probably stick with hdhomerun equipment for cable card tuners (assuming cable tv with copy once channels is the goal) because microsoft is likely to shut off program guide data for wmc 7 and 8 after the extended support for said os ends and at that point your only solution will be to swap over to hdhomerun dvr or move to another city/state that has service provided by someone who don't lock down all the major channels.
Software
Blu-ray requires aacs key updates to play new movies and most officially licensed players require you to purchase the latest version of the software to continue getting those updates after your current product goes end of life. Speaking of legit players the only major one left is Cyberlink Powerdvd as Arcsoft no longer updates or sells Total media center. (yes I know Nero and Corel make players but in my opinion they leave much to be desired. Yes there is unlicensed 3rd party solutions but due to this forum's policy that subject might be taboo here so I wont be discussing that further.